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Transmission with Energy Harvesting Nodes in Fading Wireless Channels: Optimal Policies
Omur Ozel, Kaya Tutuncuoglu, Jing Yang, Sennur Ulukus, Aylin Yener
TL;DR
The paper optimizes energy-harvesting transmission for maximizing bits by a deadline and minimizing the time to send a given amount of data. It develops offline and online policies, including directional water-filling and stochastic-dynamic-programming approaches, and identifies optimal or reduced-complexity transmission schemes.
Problem
The paper studies maximizing the number of transmitted bits by a deadline and minimizing the time required to complete transmission.
Method
The paper develops offline energy-management schemes using directional water-filling and solves online policy optimization using knowledge of event processes and stochastic methods.
Results
The paper identifies optimal offline and online transmission schemes and completes the characterization of the optimal online policy.
Takeaways & Limitations
The resulting framework covers deadline throughput maximization and transmission completion-time minimization under both offline and online knowledge.
Abstract
from arXiv · showhide
Wireless systems comprised of rechargeable nodes have a significantly prolonged lifetime and are sustainable. A distinct characteristic of these systems is the fact that the nodes can harvest energy throughout the duration in which communication takes place. As such, transmission policies of the nodes need to adapt to these harvested energy arrivals. In this paper, we consider optimization of point-to-point data transmission with an energy harvesting transmitter which has a limited battery capacity, communicating in a wireless fading channel. We consider two objectives: maximizing the throughput by a deadline, and minimizing the transmission completion time of the communication session. We optimize these objectives by controlling the time sequence of transmit powers subject to energy storage capacity and causality constraints. We, first, study optimal offline policies. We introduce a directional water-filling algorithm which provides a simple and concise interpretation of the necessary optimality conditions. We show the optimality of an adaptive directional water-filling algorithm for the throughput maximization problem. We solve the transmission completion time minimization problem by utilizing its equivalence to its throughput maximization counterpart. Next, we consider online policies. We use stochastic dynamic programming to solve for the optimal online policy that maximizes the average number of bits delivered by a deadline under stochastic fading and energy arrival processes with causal channel state feedback. We also propose near-optimal policies with reduced complexity, and numerically study their performances along with the performances of the offline and online optimal policies under various different configurations.
I. INTRODUCTION
The paper develops optimal transmission policies for an energy-harvesting transmitter with finite battery capacity in a fading channel, addressing deadline-constrained throughput and transmission completion time. It derives offline policies, including adaptive directional water-filling, and online policies using stochastic dynamic programming plus lower-complexity near-optimal algorithms.
- Problem setting: The system must adapt transmit power to random energy arrivals and fluctuating channel fades while respecting finite battery capacity and energy-causality constraints.Unused incoming energy can be wasted when the battery lacks sufficient space.
- Optimization problems: The paper optimizes two objectives: maximizing bits transmitted by deadline T and minimizing the time required to transmit B bits.
- Offline policies: For offline throughput maximization, directional water-filling accounts for energy causality by allowing stored energy to flow only toward future epochs.The static-channel solution serves as a building block for the fading case.
- Offline policies: An adaptive directional water-filling algorithm is optimal when it adjusts to both energy arrivals and channel fade levels.
- Offline policies: The completion-time problem is solved by mapping it to the deadline-constrained throughput problem using the maximum departure curve.
- Online policies: For online scheduling, stochastic dynamic programming yields an optimal power policy under statistical and causal knowledge of energy and fading variations.The paper also proposes simpler near-optimal algorithms and numerically evaluates the policies under varied system settings.
II. SYSTEM MODEL
The system models point-to-point transmission over an additive Gaussian fading channel with an energy-harvesting transmitter, causal CSI, and finite battery storage. Energy and fading changes define continuous-time epochs, while transmit power obeys energy-causality and battery-capacity constraints.
- System components: The transmitter uses separate data and energy queues, with harvested energy buffered in a battery before transmission.The battery stores at most Emax units, and processing energy is excluded from the model.
- Channel model: The channel is additive Gaussian fading, with received signal hx + n and rate 2 log (1 + hp) bits per channel use at power p.The noise is zero-mean and unit-variance Gaussian.
- Events and epochs: Fading levels and energy arrivals are stochastic processes whose changes occur at countable Poisson event times.Their rates are denoted λh and λe, and the resulting intervals are called epochs.
- Information pattern: The transmitter has causal knowledge of past energy arrivals and perfect causal feedback of the current channel fade level.This information supports online power management based on the instantaneous energy state and fading CSI.
- Power constraints: Transmit power is constrained because unavailable energy cannot be used and stored energy cannot exceed the finite battery capacity.The continuous-time model also prevents energies larger than Emax from entering the battery within a single slot.
III. MAXIMIZING THROUGHPUT IN A STATIC CHANNEL
The static-channel problem maximizes bits delivered by deadline T under energy-causality and finite-storage constraints. Its convex formulation yields a unique optimal allocation with structured power evolution.
- Problem formulation: The objective is to maximize the number of bits delivered by deadline T in a non-fading channel with offline energy-arrival knowledge.Energy arrivals occur at known times, creating N + 1 transmission epochs.
- Problem formulation: The optimization is convex because the throughput objective is concave in epoch powers and the constraints are linear.Consequently, the problem has a unique maximizer.
- Optimal allocation: With infinite battery capacity, optimal power levels form a monotonically increasing sequence.Unused energy can be carried from the current epoch to future epochs, producing nondecreasing power.
- Optimal allocation: When energy is transferred to a future epoch, adjacent power levels remain equal; power changes occur only as increases when the corresponding causality constraint binds.This characterizes the structure behind the directional water-filling interpretation.
- Finite storage: Finite Emax limits energy transfer from current to future epochs to Emax − Ei, so power equalization may be restricted.The resulting allocation can lose global monotonicity when battery-capacity constraints bind.
A. Directional Water-Filling Algorithm
Directional water-filling interprets optimal power allocation as energy flowing forward across epochs, never backward, while battery capacity limits how much energy can pass between them.
- Water-filling interpretation: Right permeable taps allow energy to flow only from earlier epochs to later epochs, reflecting energy-causality.Water levels therefore need not be equalized across an impermeable direction.
- Water-filling interpretation: The directional water-filling algorithm represents each epoch as a rectangle whose water level corresponds to energy divided by epoch duration.If E units fill a rectangle of width L, the water level is E/L.
- Two-epoch example: With sufficiently large Emax, energy can flow from epoch 1 to epoch 2 to equalize their water levels, but never in reverse.The two-epoch construction illustrates the directional restriction directly.
- Finite storage: With finite Emax, a tap transfers at most Emax − E1 from epoch 1 to epoch 2.Allowing more transfer would exceed the next epoch’s battery capacity and cause energy overflow.
IV. MAXIMIZING THROUGHPUT IN A FADING CHANNEL
The fading-channel throughput problem extends the offline allocation across epochs formed by both fading changes and energy arrivals. Its optimal water level is nondecreasing with infinite storage, while finite storage limits inter-epoch transfer and can disrupt monotonicity.
- Problem formulation: The fading-channel model forms M + N + 1 epochs from M channel-state changes and N energy arrivals, with constant transmit power within each epoch.An epoch receives energy Ej when an energy event occurs and zero energy when only the fade changes.
- Problem formulation: The fading-channel optimization remains a convex problem with a unique optimal solution under energy-causality and battery-capacity constraints.Some optimal powers may be zero depending on the channel fading state.
- Infinite storage: With infinite battery capacity, the optimal water level νi is monotonically increasing across epochs.If energy transfers from epoch i to i + 1, then νi = νi+1.
- Infinite storage: Between consecutive energy arrivals, all intervening epochs have equal water levels because injected energy spreads freely across them.There is no energy-transfer wall between these epochs.
- Finite storage: Finite Emax limits transferred energy to Emax − Ein(i), so water-level monotonicity may fail across energy-arrival boundaries.Between energy arrivals, levels remain equalized, but the next level may be higher or lower after a new arrival.
A. Directional Water-Filling Algorithm
The directional water-filling algorithm allocates energy across fading-channel epochs while enforcing causality, battery capacity, and channel-dependent transmission choices.
- No walls separate epochs solely because of fading-level changes; water levels are determined by directional water-filling.The resulting water levels are substituted into the power-allocation expression to obtain optimal powers.
- In the 12-epoch example, three energy arrivals occur during transmission in addition to the initial energy.The algorithm is illustrated for epochs labeled L1 through L12.
- Power is zero in epochs 1 and 3 because their channel gains are too high, while energy equalizes across selected later epochs.Energy equalizes in epochs 2, 4, and 5, and again across epochs 8 through 12.
- Energy arriving at epoch 6 cannot flow left, and excess energy in epochs 6 and 7 cannot flow right because of the Emax constraint.The right-permeable tap between epochs 7 and 8 enforces the storage limitation.
- Right-permeable taps allow energy to flow only toward later epochs, enforcing energy-arrival causality.The algorithm uses taps that limit transferred water to at most Emax at each wall.
V. TRANSMISSION COMPLETION TIME MINIMIZATION IN FADING CHANNEL
The paper addresses completion-time minimization for transmitting B bits in an energy-harvesting fading channel by introducing the maximum departure curve and characterizing its properties.
- The completion-time problem minimizes the time needed to transmit B bits under energy harvesting and fading-channel conditions.The setting includes a finite battery constraint and builds on related non-fading and battery-limited formulations.
- The maximum departure curve D(T) maps a deadline and given energy and fading sequences to the maximum number of served bits.It connects completion-time minimization to the throughput-maximization problem.
- D(T) is monotonically increasing and continuous, but its derivative can be nondifferentiable at energy arrivals and fading changes.The derivative represents the rate of energy transfer from past into the future at time T.
- Right-permeable taps prevent water from flowing left after energy arrivals, producing derivative discontinuities in D(T).Fading changes alter water-level evolution through the reciprocal channel gains, while energy arrivals can create abrupt changes in transfer behavior.
- With no fading or energy arrivals, the optimal policy is constant power and D(T) is continuous and monotonically increasing.Under a finite battery, the long-time asymptote can be strictly smaller than with unlimited battery capacity.
- Battery capacity can create additional discontinuities in D′(T) beyond those caused by fading changes and energy arrivals.This is explicitly identified as a consequence of the Emax constraint.
B. Solution of the Transmission Completion Time Minimization Problem in a Fading Channel
The minimum completion time is obtained from the maximum departure curve: it is the earliest time at which the curve reaches the required payload B.
- Transmission completion-time minimization is closely related to maximizing the number of bits sent by a deadline.If D(T) is less than B, completing B bits by T is impossible.
- The optimal completion time T* satisfies B = D(T*).This equality characterizes the boundary between deadlines that cannot and can support transmission of B bits.
- Theorem 3 gives T* = min{t ∈ MB}, where MB = {t : B = D(t)}.Because D(T) is continuous, the relevant minimizing time exists and is unique.
VI. ONLINE TRANSMISSION POLICIES
The online problem maximizes expected bits delivered by deadline T using only causal energy and channel information. A stochastic dynamic-programming solution yields the optimal policy, while recursive discretization and lookup-table implementation make it operational.
- The online objective is to maximize the number of bits sent by deadline T using causal energy-arrival and channel-fade information.Energy arrivals and fading are modeled as marked Poisson processes with specified rates and densities.
- The system state consists of fade level h and battery energy e, and g(e,h,t) specifies the transmit power at time t.Admissible policies are nonnegative, transmit no power at zero battery energy, and leave zero energy at the deadline.
- The value function is the supremum of expected throughput over admissible policies.The throughput is the expected number of bits sent under policy g by time t.
- The optimal online policy g* solves the dynamic-programming value equation.The policy is computed recursively after quantizing time with a sufficiently small δ-skeleton.
- The continuous-time dynamic-programming procedure produces the optimal online policy, which can be stored as a lookup table.At each time, the transmitter uses current fade feedback and battery energy to select g*(e(t),h(t),t).
B. Other Online Policies
The paper proposes reduced-complexity online transmission policies that react to energy and fading events, trading some performance for lower computation and statistical requirements.
- These simpler policies reduce computation and feedback, but the constant water level policy is strictly suboptimal in the time-constrained setting.
- Event-based policies recalculate transmit power when fading changes or energy arrives, subject to battery availability and the Emax constraint.
- Constant Water Level Policy: Constant water level uses causal fading feedback and requires knowledge of the mean energy arrival rate and full fading statistics.
- Energy Adaptive Water-Filling: The energy-adaptive water-filling variant recalculates a cutoff fade level and power after each energy arrival using the current battery energy.
- Time-Energy Adaptive Water-Filling: Time-energy adaptive water filling additionally adapts power to the remaining time before the deadline.
3) Time-Energy Adaptive Water-Filling:
The time-energy adaptive water-filling policy is evaluated against optimal and other online policies across fading, recharge, battery, and deadline settings. It performs well at low recharge rates but can lose efficiency as recharge or deadline duration increases because limited storage causes overflows.
- The simulations compare offline and online optimal policies with water-filling-based suboptimal policies that react to energy arrivals and fading changes.
- The offline optimum remains below the noncausal upper bound because energy causality prevents energy from being spread evenly across the entire deadline.
- Under Nakagami fading with m = 3, the energy-adaptive water-filling policy performs worse than constant and time-energy adaptive water filling.
- As deadline T increases, the constant water level policy approaches the optimal online policy, reducing the importance of time awareness.
- Time-energy adaptive water filling degrades for longer deadlines because spreading energy over long intervals lowers power, increases battery accumulation, and causes overflows.
VIII. CONCLUSIONS
The paper develops energy-management schemes for fading channels with finite-capacity rechargeable batteries. It solves offline throughput and completion-time problems, derives an online dynamic-programming policy, and evaluates these algorithms numerically.
- The paper studies maximizing bits delivered by a deadline and minimizing the time needed to transmit a specified amount of data.
- Numerical results report the performance of the offline and online algorithms under different Nakagami fading configurations and deadlines.
- Offline throughput optimization uses directional water filling, while completion-time minimization is mapped to throughput maximization through the maximum departure curve.
- The optimal online deadline-constrained policy is obtained using continuous-time dynamic programming under online knowledge of events.